Average Error: 33.6 → 10.4
Time: 20.8s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.136683434005781 \cdot 10^{-32}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 2.927598127340643 \cdot 10^{+124}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -3.136683434005781 \cdot 10^{-32}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le 2.927598127340643 \cdot 10^{+124}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\end{array}
double f(double a, double b, double c) {
        double r3229451 = b;
        double r3229452 = -r3229451;
        double r3229453 = r3229451 * r3229451;
        double r3229454 = 4.0;
        double r3229455 = a;
        double r3229456 = c;
        double r3229457 = r3229455 * r3229456;
        double r3229458 = r3229454 * r3229457;
        double r3229459 = r3229453 - r3229458;
        double r3229460 = sqrt(r3229459);
        double r3229461 = r3229452 - r3229460;
        double r3229462 = 2.0;
        double r3229463 = r3229462 * r3229455;
        double r3229464 = r3229461 / r3229463;
        return r3229464;
}
↓
double f(double a, double b, double c) {
        double r3229465 = b;
        double r3229466 = -3.136683434005781e-32;
        bool r3229467 = r3229465 <= r3229466;
        double r3229468 = c;
        double r3229469 = r3229468 / r3229465;
        double r3229470 = -r3229469;
        double r3229471 = 2.927598127340643e+124;
        bool r3229472 = r3229465 <= r3229471;
        double r3229473 = -r3229465;
        double r3229474 = -4.0;
        double r3229475 = a;
        double r3229476 = r3229475 * r3229468;
        double r3229477 = r3229465 * r3229465;
        double r3229478 = fma(r3229474, r3229476, r3229477);
        double r3229479 = sqrt(r3229478);
        double r3229480 = r3229473 - r3229479;
        double r3229481 = 2.0;
        double r3229482 = r3229481 * r3229475;
        double r3229483 = r3229480 / r3229482;
        double r3229484 = r3229465 / r3229475;
        double r3229485 = r3229469 - r3229484;
        double r3229486 = r3229472 ? r3229483 : r3229485;
        double r3229487 = r3229467 ? r3229470 : r3229486;
        return r3229487;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.6
Target20.8
Herbie10.4
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -3.136683434005781e-32

    1. Initial program 53.4

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

    2. Taylor expanded around -inf 7.3

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]

    3. Simplified7.3

      \[\leadsto \color{blue}{-\frac{c}{b}}\]

    if -3.136683434005781e-32 < b < 2.927598127340643e+124

    1. Initial program 14.7

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

    2. Taylor expanded around 0 14.7

      \[\leadsto \frac{\left(-b\right) - \sqrt{\color{blue}{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]

    3. Simplified14.7

      \[\leadsto \frac{\left(-b\right) - \sqrt{\color{blue}{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}}{2 \cdot a}\]

    if 2.927598127340643e+124 < b

    1. Initial program 50.6

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

    2. Taylor expanded around inf 2.9

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.136683434005781 \cdot 10^{-32}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 2.927598127340643 \cdot 10^{+124}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r2)"

  :herbie-target
  (if (< b 0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))